The basic problem in this paper is that of detemnining the geometry of an arbitrary multiply connected bounded region in R 3 together with the mixed boundary conditions, from the complete knowledge of the eigenvalues { λ j } j = 1 ∞ for the negative Laplacian, using the asymptotic expansion of the spectral function θ ( t ) = ∑ j = 1 ∞ exp ( − t λ j ) as t → 0 .